\begindata{text, 204354} \textdsversion{11} \template{default} \define{invisible } \define{zserif menu:[Font,Andy] attr:[FontFamily Andy Int 0]} \define{zsans menu:[Font,AndySans] attr:[FontFamily AndySans Int 0]} \define{zfixed menu:[Font,AndyFixed] attr:[FontFamily AndyFixed Int 0]} \define{zsymbol menu:[Font,AndySymbol] attr:[FontFamily AndySymbol Int 0]} $syntaxlevel 1 *Module for Linear Regression \flushleft{define }\flushleft{\tim{i: i,n,flag,j,l,sf(1:3),cc,xcalc,n0,cc0,xcalc0,xcol,ycol,flag2 f: Sx,Sy,Sxy,Sx2,st,s(0:30),a1,b1,a,b,x1,y1,lastx,lasty,m,int,d,eb,em,xave,yave,sdx,sdyx,clmp,clmp1,x0,studT,k,k1,sig,t(-40:40) font zsans,25 fine 500,400 rescale TRUE,TRUE,TRUE,TRUE unit genesis calc xcalc:=4 $$ point of limit sampling cc:=95 $$ confidence coeffcient. studT:=1.71 $$ T distribution value n:=25 $$ no. sampled points jump cover unit cover }}\flushleft{menu Title: pre0 }menu Least Squares Fit: pre2 menu Least Squares Fit Sample:pre2a menu Confidence Limits Assumptions: pre3 menu Confidence Limits for Slope: pre4 menu Confidence Limits for Slope Sample: pre4aa menu Confidence Limits for Mean:pre4half menu Confidence Limits for Mean Sample:pre4a menu Confidence Limits Conclusion: pre5 menu Prediction Limits: pre6 menu Prediction Limits Sample:pre6a menu Conclusion: pre7 menu Supplemental Information: pre8 next pg2 text \center{Tutorial on: }\bold{\bigger{\center{Linear Regression, Confidence Limits for a Population Mean, and Prediction Limits for One More Observation}}}\center{ Frank McClintock, Terence Chow, Louis L. Bucciarelli Spring 1991 This module is to provide educational information on statistical analysis and to illustrate concepts in statistical analysis. \\ }do nextscr jump pg2 unit pg2 next sample back cover text Linear regression is used to fit a linear relation between two variables, from pairs of corresponding data points. The coefficients \italic{\tim{a}}\tim{ and }\italic{\tim{b}}\tim{ }of a line \italic{\tim{y = a + bx}} are chosen to minimize the sum of the squares of \italic{\tim{y}}-errors between the data points and the line; that is, we choose \italic{a} and \italic{b} to minimize: \flushleft{ }\italic{\center{\symb{S}}}\center{(}\italic{\center{y}}\italic{\subscript{\center{\tim{i}}}}\italic{\center{ - }}\center{(}\italic{\center{a + bx}}\italic{\subscript{\center{\tim{i}}}}\center{))}\superscript{\center{2}}\center{. }This line is referred to as a \italic{least squares fit}. A least squares fit is performed on a \italic{sample}, a finite collection of data points. A sample is a subset of the \italic{population}, an infinite set of data. \flushleft{\\ do nextscr jump sample unit pg3 next pg4 back sample text }Although the infinite population itself cannot be found from a finite sample, \italic{confidence limits} for the slope and for the mean value of the population at any \italic{x} can be found from the sample. First however, make four assumptions about the infinite population and the relative errors: 1. at any \italic{x}, the values of \italic{y} are normally distributed about the mean value of \italic{y}, 2. the mean value of\italic{ y} is a linear function of \italic{x}, 3. the standard deviation of \italic{y}\italic{\subscript{i}} about the mean line is constant, independent of \italic{x}, and 4. there is no error in the values of the independent variable, \italic{x}\italic{\subscript{i}}. \flushleft{ \\ do nextscr jump pg4 unit pg4 next sample3 back pg3 text }The \italic{confidence limits for the slope of the population} have the form:\tim{ \\ font zsans,20 calc i:=zwherey+40} at 15,i text confidence limit for population slope\tim{ }\flushleft{\tim{\\ at zwherex,i+10 write = }}\flushleft{sample slope }\flushleft{\tim{+}}\flushleft{/- at zwherex,i text half the confidence interval (}\italic{\flushleft{c%}}\flushleft{) \\ font at 0,zwherey+40 text }The \italic{confidence coefficient} \italic{c} is defined such that for repeated sampling from the population, and construction from each sample of the \italic{c%} confidence limits for some population characteristic, \italic{c%} of the time the population characteristic will lie within those limits.\tim{ \\ }\flushleft{\tim{do nextscr jump sample3 unit pg4half next sample2 back sample3 text }}\flushleft{The }\italic{\flushleft{confidence limits for the mean of the population at any x}}\flushleft{ have a similar form, but change with x:}\flushleft{\tim{ \\ font zsans,20 calc i:=zwherey+40 at 15,i text confidence limits \\ at zwherex,i+10 write = at zwherex,i write least squares mean value at }}\italic{\flushleft{\tim{x }}}at zwherex,i+10 \flushleft{write }\flushleft{\tim{+/- half the confidence interval at }}\italic{\flushleft{\tim{x}}}\flushleft{\tim{ (}}\italic{\flushleft{\tim{c}}}\flushleft{\tim{%) at 0,i+20 write for mean font at 0,zwherey+40 text }}This equation gives a pair of curves, the (\italic{c%}) confidence limits, useful in judging where the mean of the population at any \italic{x} may lie.\tim{ }\flushleft{\tim{\\ }}\flushleft{at 205,187 write }\flushleft{\tim{ do nextscr jump sample2 unit pg5 next pg6 back sample2 text }}Again, the meaning of the confidence coefficient of 95%, is that 95% of the times a sample is drawn and confidence limits are constructed, these limits \italic{do} include the mean population at that \italic{x}, and 5% of the time they do not. \\\tim{ }\flushleft{\tim{do nextscr jump pg6 unit pg6 next sample4 back pg5 text }}\italic{Prediction limits for one more observation} can also be constructed. For a prediction coefficient \italic{c%} and a specific location \italic{x}, the prediction limits provide an interval in \italic{y}. \italic{c%} of the times a sample is drawn and one more data point is observed at that location \italic{x}, the point will lie within the prediction interval; (100 - \italic{c})\italic{%} of the time, it will lie outside of the interval. the prediction interval for more observation is greater than the corresponding confidence interval because not only is the population mean unknown, but also the \italic{y} value of the single sample.\tim{ }\flushleft{\tim{\\ do nextscr jump sample4 unit pg7 next userchoose back sample4 text }}Confidence limits are useful in determining the relationship of certain characteristics to others for very large amounts of data. The prediction limits are important in cases where one experiment or one device is to be produced, and an appropriate range of its characteristics \italic{must} be known. \\\tim{ }\flushleft{\tim{do nextscr jump userchoose unit bib next userchoose back pg7 text }}Further Reading: McClintock, Frank A., Statistical Estimation: Linear Regression and Single Variable, Research Memo 274 (1989). McClintock, Frank A., \italic{Small Sample Statistics}, 2.671 Course Notes (1972). Dixon, W. J. and F. J. Massey, Jr.: "Introduction to Statistical Analysis," McGraw-Hill Book Company, Inc. New York 1951. \\\tim{ }\flushleft{\tim{do nextscr jump userchoose unit programcode }}\tim{text }Program code information: \\\tim{ }\flushleft{\tim{font zsans,15 text}}\flushleft{\helv{ a = true intercept = 3 Sx = Sx b = true slope = .7 Sy = Sy n = number of sampled points = 25 Sxy = Sx*y sig = standard deviation of sampled points =1 Sx2 = Sx}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{ studT = t distribution value (95%) = 1.71 a1 = least squares intercept = (Sx2 * Sy - Sx * Sxy)/(n * Sx2 - (Sx)}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{) b1 = least squares slope = (n * Sxy - Sy * Sx)/(n * Sx2 - (Sx)}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{) xave = average x = Sx/n yave = average y = Sy/n sdx = standard deviation of x about their mean = [1/(n - 1) * S(x0 - xave)}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{]}}\superscript{\flushleft{\helv{.5}}}\flushleft{\helv{, 0 < x0 < 8 sdyx = standard deviation of y-data about least squares line = \{1/(n - 2) * S[y0 - (a1 + b1 * x0)]}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{\}}}\superscript{\flushleft{\helv{.5}}}\flushleft{\helv{, y0 = a1 + b1 * x0, 0 < x0 < 8 b = confidence limits for slope = b1 +/- studT * sdyx/sdx * (n-1)}}\superscript{\flushleft{\helv{-.5 }}}\flushleft{\helv{m}}\subscript{\flushleft{\helv{y}}}\flushleft{\helv{(x) = confidence limits for mean = yave + b1 * (x0 - xave) +/- studT * sdyx * \{1/n + (x0 - xave)}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{/[(n - 1) * xave}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{]\}}}\superscript{\flushleft{\helv{.5}}}\flushleft{\helv{ y}}\subscript{\flushleft{\helv{n+1}}}\flushleft{\helv{ = prediction limits for one more observation = yave + b1 * (x0 - xave) +/- studT * sdyx * \{1 + 1/n + (x0 - xave)}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{/[(n - 1) * xave}}\superscript{\flushleft{\helv{2}}}\flushleft{\helv{]\}}}\superscript{\flushleft{\helv{.5 \\}}}\superscript{\flushleft{\tim{ }}}font do nextscr2 jump userchoose unit nextscr2 box 25,375;475,396;3 text 1,372 \center{Click to return to supplemental information page }\\ loop pause keys=touch \flushleft{ }\flushleft{\tim{if (ztouchx>24)&(ztouchx<476)&(ztouchy>374)&(ztouchy<397) outloop endif endloop mode xor box 25,375;475,396;-12 pause 1 erase mode write}}\superscript{\flushleft{\tim{ }}}\flushleft{\tim{ unit nextscr box 125,375;375,396;3 text 1,373 }}\center{Click here to continue}\center{\tim{ }}\flushleft{\tim{\\ loop pause keys=touch if (ztouchx>124)&(ztouchx<376)&(ztouchy>374)&(ztouchy<397) outloop endif endloop mode xor box 125,375;375,396;-12 pause 1 erase mode write unit pre0 jump cover unit pre2 jump pg2 unit pre3 jump pg3 unit pre4 jump pg4 unit pre4half jump pg4half unit pre5 jump pg5 unit pre6 jump pg6 unit pre7 jump pg7 unit pre8 jump userchoose unit pre2a jump sample unit pre4aa jump sample3 unit pre4a jump sample2 unit pre6a jump sample4 unit graph gorigin 225,250 $$set up and dimension graph axes 265,240 scalex 8 scaley 11 labelx 1,.5 labely 1,.5 at 468,275 $$label axes }}\tim{write }x\tim{ }\flushleft{\tim{at 185,12 }}\tim{write }y\tim{ }\flushleft{\tim{ unit points at 15,0 }}\tim{write }x\tim{ }\flushleft{\tim{at 55,0 }}\tim{write }y\subscript{i}\tim{ }\flushleft{\tim{calc a:=3 $$a=y-intercept of line b:=.7 $$b=slope of line sig:=1 $$std dev of sampled points calc st :=8/n $$st=step between sampled points}}\superscript{\flushleft{\tim{ do pickpoints font zsans,20 }}}\flushleft{\tim{loop i :=1,n at 0,i*13+10 $$show point and coordinates of point show st*i,3 at 50,i*13+10 show s(i),3 gat st*i,s(i) gcircle .05 endloop font unit pickpoints calc Sx:=0 $$Sx=}}\flushleft{\symb{S}}\flushleft{\tim{x Sy:=0 $$Sy=}}\flushleft{\symb{S}}\flushleft{\tim{y Sxy:=0 $$Sxy=}}\flushleft{\symb{S}}\flushleft{\tim{x*y Sx2:=0 $$Sx2=}}\flushleft{\symb{S}}\flushleft{\tim{x}}\superscript{\flushleft{\tim{2}}}\flushleft{\tim{ loop i:=1,n $$seect n random points do random calc s(i):=k1+b*st*i+a Sy :=Sy+s(i) $$sum up points Sx :=Sx+st*i Sx2 :=Sx2+(st*i)^2 Sxy :=Sxy+s(i)*st*i endloop unit random randu k,1000 calc k:=k/1000 if k<=.0797 calc k1:=.1*sig elseif k<=.1585 calc k1:=.2*sig elseif k<=.2358 calc k1:=.3*sig elseif k<=.3108 calc k1:=.4*sig elseif k<=.3829 calc k1:=.5*sig elseif k<=.4515 calc k1:=.6*sig elseif k<=.5161 calc k1:=.7*sig elseif k<=.5763 calc k1:=.8*sig elseif k<=.6319 calc k1:=.9*sig elseif k<=.6827 calc k1:=sig elseif k<=.7287 calc k1:=1.1*sig elseif k<=.7699 calc k1:=1.2*sig elseif k<=.8064 calc k1:=1.3*sig elseif k<=.8385 calc k1:=1.4*sig elseif k<=.8664 calc k1:=1.5*sig elseif k<=.8904 calc k1:=1.6*sig elseif k<=.9109 calc k1:=1.7*sig elseif k<=.9281 calc k1:=1.8*sig elseif k<=.9426 calc k1:=1.9*sig elseif k<=.9545 calc k1:=2*sig elseif k<=.9643 calc k1:=2.1*sig elseif k<=.9722 calc k1:=2.2*sig elseif k<=.9786 calc k1:=2.3*sig elseif k<=.9836 calc k1:=2.4*sig elseif k<=.9876 calc k1:=2.5*sig elseif k<=.9907 calc k1:=2.6*sig elseif k<=.9931 calc k1:=2.7*sig elseif k<=.9949 calc k1:=2.8*sig elseif k<=.9963 calc k1:=2.9*sig elseif k<=.9973 calc k1:=3*sig else calc k1:=3.5*sig endif randu k,2 calc k1:=k1*(-1)^k unit berase erase 0,300;500,400 unit average calc xave:=Sx/n yave:=Sy/n calc a1 :=(Sx2*Sy-Sx*Sxy)/(n*Sx2-Sx^2) calc b1 :=(n*Sxy-Sy*Sx)/(n*Sx2-Sx^2) em:=abs(m-b1) $$calculate errors eb:=abs(int-a1) loop i:=1,n calc sdx:=sdx+(st*i-xave)^2 sdyx:=sdyx+(s(i)-(a1+b1*st*i))^2 endloop calc sdx:=(sdx/(n-1))^.5 sdyx:=(sdyx/(n-2))^.5 unit drawline loop if flag2=0 erase 100,300;500,400 at 100,300 }}\tim{ write }Hold the mouse button down and drag a line you think is the least squares fit. elseif flag2=2 erase 100,300;500,400 at 100,300 write Try again.\tim{ }\flushleft{\tim{ endif loop pause keys=touch if (zgtouchx>0)&(zgtouchx<8)&(zgtouchy>0)&(zgtouchy<11) outloop endif endloop calc x1:=lastx:=zgtouchx y1:=lasty:=zgtouchy gdot x1,y1 mode xor loop pause keys=touch(left:move, up) outloop zkey=zk(left:up) if (zgtouchx>0)&(zgtouchx<8)&(zgtouchy>0)&(zgtouchy<11) gdraw x1,y1;lastx,lasty gdraw x1,y1;zgtouchx,zgtouchy calc lastx:=zgtouchx lasty:=zgtouchy endif endloop *analyze line drawn if x1=lastx gdraw x1,y1;lastx,lasty gdot x1,x1 calc flag2:=1 erase 100,300;500,400 if y1=lasty at 100,300 }}\tim{ write }Please draw a line.\tim{ }\flushleft{\tim{ else at 100,300 }}\tim{ write }Please draw a nonvertical line.\tim{ }\flushleft{\tim{ endif else outloop endif endloop unit sample $$ex of least squares fit next pg3 back pg2 do graph calc flag2:=0 text 0,300 }}In the following few screens, you will see an example of the application of the least squares fit. Click the mouse button to see a sample of points. \\\tim{ }\flushleft{\tim{pause keys=all,touch do berase }}\subscript{\flushleft{\tim{do points}}}\flushleft{\tim{ loop do drawline erase 100,300;500,400 text 100,300 }}Are you satisfied with your line?\tim{ }\flushleft{\tim{\\ box 180,375;220,396;3 at 185,371 }}\tim{ write }yes\tim{ }\flushleft{\tim{ box 250,375;290,396;3 at 258,371 }}\tim{ write }no\tim{ }\flushleft{\tim{ loop calc flag:=0 pause keys=touch if (ztouchx>179)&(ztouchx<221)&(ztouchy>374)&(ztouchy<397) calc flag:=1 box 180,375;220,396;-16 pause 1 outloop elseif (ztouchx>249)&(ztouchx<291)&(ztouchy>374)&(ztouchy<397) box 250,375;290,396;-16 pause 1 outloop endif endloop if flag=1 outloop else gdraw x1,y1;lastx,lasty gdot x1,y1 endif calc flag2:=2 endloop calc m:=(y1-lasty)/(x1-lastx) $$slope? int:=y1-m*x1 $$intercept? erase 100,300;500,400 text 100,300 }}Now watch as the computer plots the actual least squares fit.\tim{ }\flushleft{\tim{\\ pause 2 do average loop x0:=0,8,.1 gdot x0,a1+x0*b1 endloop do berase erase 0,0;150,400 font zsans,20 at 0,0 }}write Your slope: <|showt,m,2,2|> Actual slope: <|showt,b1,2,2|> Slope error: <|showt,em,2,2|> Your y-intercept: <|showt,int,2,2|> Actual y-intercept: <|showt,a1,2,2|> y-intercept error: <|showt,eb,2,2|> Actual equation: y = <|showt,a1,2,2|> + <|showt,b1,2,2|> * x Your equation: y = <|showt,int,2,2|> + <|showt,m,2,2|> * x font at 20,255 if (em<.1)&(eb<.1) write FANTASTIC! elseif (em<.3)&(eb<.3) write GREAT! elseif (em<.5)&(eb<.5) write GOOD! elseif (em<.7)&(eb<.7) write NOT BAD! elseif (em<1)&(eb<1) write OKAY! elseif (em<1.5)&(eb<1.5) write PRETTY GOOD! else write OOPS! endif text 0,300 The dashed line is the least squares fit. On the left side of the screen, you can compare your line to the least squares fit of this sample. Please click to continue. \\\tim{ }\flushleft{\tim{pause keys=touch pause 1 jump pg3 unit sample2 $$NEW example of confidence limits for the mean of the population next pg5 back pg4half mode xor do graph text 0,300 }}In the following few screens, you will see an example of confidence limits for the mean of the population. Click to see a sample of points. \\\tim{ }\flushleft{\tim{pause keys=touch do berase do points do exprelim do berase at 0,300 }}\tim{write }Now click to see if the <|s,cc|>% confidence limits include the true mean at \italic{x} = <|s,xcalc|>.\tim{ }\flushleft{\tim{pause keys=touch loop x0:=0,30,3 dot x0,180 endloop font zsans,20 at 35,170 }}\tim{write }confidence limits\tim{ }\flushleft{\tim{font calc flag:=0 loop x0:=0,8,.1 calc clmp:=yave+b1*(x0-xave)+studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 clmp1:=yave+b1*(x0-xave)-studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 gdot x0,clmp gdot x0,clmp1 endloop calc x0:=xcalc clmp:=yave+b1*(x0-xave)+studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 clmp1:=yave+b1*(x0-xave)-studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 if (clmpa+b*x0) calc flag:=1 endif gvector xcalc,.5;xcalc,a+b*xcalc-2;7 gvector xcalc,11;xcalc,a+b*xcalc+2;7 do berase at 0,300 if flag=1 }}\tim{ }write The confidence limits do not include the true mean at \italic{x} = <|s,xcalc|>! else write The confidence limits do include the true mean at \italic{x} = <|s,xcalc|>! endif\tim{ }\flushleft{\tim{pause 2 at 0,350 }}\tim{write }Please click to continue.\tim{ }\flushleft{\tim{pause keys=all,touch erase 0,0;150,400 do berase text 0,300 }}Now we will test the confidence limits of other samples. Click to indicate how many samples you wish to see. \\\tim{ }\flushleft{\tim{do boxes loop l:=1,j do sample2more if flag=1 calc sf(1):=sf(1)+1 else calc sf(2):=sf(2)+1 endif erase 110,0;150,70 at 110,0 show sf(2) at 110,40 show sf(1) endloop calc sf(3):=100*sf(2)/(sf(1)+sf(2)) do berase at 0,300 }}\tim{write }For these samplings, there were <|s,sf(2)|> successes and <|s,sf(1)|> failures for a success rate of <|s,sf(3)|>%.\tim{ }\flushleft{\tim{pause 1 do nextscr jump pg5 unit sample2more loop i:=1,n gat st*i,s(i) gcircle .05 endloop loop x0:=0,8,.1 calc clmp:=yave+b1*(x0-xave)+studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 clmp1:=yave+b1*(x0-xave)-studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 gdot x0,clmp gdot x0,clmp1 endloop do pickpoints loop i:=1,n gat st*i,s(i) gcircle .05 endloop do average calc flag:=0 loop x0:=0,8,.1 calc clmp:=yave+b1*(x0-xave)+studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 clmp1:=yave+b1*(x0-xave)-studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 gdot x0,clmp gdot x0,clmp1 endloop calc x0:=xcalc clmp:=yave+b1*(x0-xave)+studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 clmp1:=yave+b1*(x0-xave)-studT*sdyx*(1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 if (clmpa+b*x0) calc flag:=1 endif pause .5 unit sample3more loop i:=1,n gat st*i,s(i) gcircle .05 endloop loop x0:=0,8,.1 gdot x0,clmp*x0+a gdot x0,clmp1*x0+a endloop do pickpoints loop i:=1,n gat st*i,s(i) gcircle .05 endloop do average calc flag:=0 calc clmp:=b1+studT*sdyx/sdx*(n-1)^-.5 clmp1:=b1-studT*sdyx/sdx*(n-1)^-.5 loop x0:=0,8,.1 gdot x0,clmp*x0+a gdot x0,clmp1*x0+a endloop if (clmpb) calc flag:=1 endif pause .5 unit boxes mode xor box 100,375;140,396;3 box 150,375;190,396;3 box 200,375;240,396;3 box 250,375;290,396;3 box 300,375;340,396;3 at 115,373 }}\tim{write }1\tim{ }\flushleft{\tim{at 158,373 }}\tim{write }10\tim{ }\flushleft{\tim{at 208,373 }}\tim{write }25\tim{ }\flushleft{\tim{at 258,373 }}\tim{write }50\tim{ }\flushleft{\tim{at 304,373 }}\tim{write }100\tim{ }\flushleft{\tim{loop pause keys=touch if (ztouchx>99)&(ztouchx<141)&(ztouchy>374)&(ztouchy<397) box 100,375;140,396;-16 calc j:=1 outloop elseif (ztouchx>149)&(ztouchx<191)&(ztouchy>374)&(ztouchy<397) box 150,375;190,396;-16 calc j:=10 outloop elseif (ztouchx>199)&(ztouchx<241)&(ztouchy>374)&(ztouchy<397) box 200,375;240,396;-16 calc j:=25 outloop elseif (ztouchx>249)&(ztouchx<291)&(ztouchy>374)&(ztouchy<397) box 250,375;290,396;-16 calc j:=50 outloop elseif (ztouchx>299)&(ztouchx<341)&(ztouchy>374)&(ztouchy<397) box 300,375;340,396;-16 calc j:=100 outloop endif endloop loop x0:=0,8,.3 gdraw x0,a1+x0*b1;x0+.2,a1+(x0+.2)*b1 endloop do berase erase 0,0;150,400 at 0,0 }}\tim{write }Successes:\tim{ }\flushleft{\tim{at 0,40 }}\tim{write }Failures:\tim{ }\flushleft{\tim{calc sf(1):=sf(2):=0 }}\tim{at 0,300} write Watch at the top left as the number of successes and failures is recorded.\tim{ }\flushleft{\tim{pause 2 unit sample4 $$NEW example of prediction limits next pg7 back pg6 mode xor do graph text 0,300 }}In the following few screens, you will see an example of prediction limits for one more observation. Click to see a sample of points. \\\tim{ }\flushleft{\tim{pause keys=all,touch do berase do points do exprelim do berase at 0,300 }}\tim{write }Now click to see the <|s,cc|>% prediction limits for one more observation.\tim{ }\flushleft{\tim{pause keys=all,touch loop x0:=0,30,3 dot x0,180 endloop font zsans,20 at 35,170 }}\tim{write }prediction limits\tim{ }\flushleft{\tim{font calc studT:=1.71 flag:=0 loop x0:=0,8,.1 calc clmp:=yave+b1*(x0-xave)+studT*sdyx*(1+1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 clmp1:=yave+b1*(x0-xave)-studT*sdyx*(1+1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 gdot x0,clmp gdot x0,clmp1 endloop pause 1 do berase erase 0,0;150,400 text 0,300 }}Now we will test the prediction limits by sampling points at \italic{x} = <|s,xcalc|>. Click to indicate how many samples you wish to see. \\\tim{ }\flushleft{\tim{do boxes loop i:=1,n gat st*i,s(i) gcircle .05 endloop calc x0:=xcalc clmp:=yave+b1*(x0-xave)+studT*sdyx*(1+1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 clmp1:=yave+b1*(x0-xave)-studT*sdyx*(1+1/n+(x0-xave)^2/((n-1)*sdx^2))^.5 loop l:=1,j do random calc s(1):=k1+b*xcalc+a gat xcalc,s(1) gcircle .05 do average calc flag:=0 if (clmps(1)) calc flag:=1 endif if flag=1 calc sf(1):=sf(1)+1 else calc sf(2):=sf(2)+1 endif erase 110,0;150,70 at 110,0 show sf(2) at 110,40 show sf(1) pause 1 gat xcalc,s(1) gcircle .05 endloop calc sf(3):=100*sf(2)/(sf(1)+sf(2)) do berase at 0,300 }}\tim{write }For these samplings, there were <|s,sf(2)|> successes and <|s,sf(1)|> failures for a success rate of <|s,sf(3)|>%.\tim{ }\flushleft{\tim{pause 1 do nextscr jump pg7 unit exprelim at 100,300 }}\tim{write }Now click to see the true mean line.\tim{ }\flushleft{\tim{pause keys=all,touch pause .5 erase 0,0;150,400 do average gdraw 0,a;8,a+8*b gdraw 0,a-.03;8,a+8*b-.05 box 0,100;30,102 font zsans,20 at 35,90 }}\tim{write }true mean line\tim{ }\flushleft{\tim{font do berase at 0,300 }}\tim{write }Click again to see the least squares fit.\tim{ }\flushleft{\tim{pause keys=all,touch pause .5 loop x0:=0,8,.3 gdraw x0,a1+x0*b1;x0+.2,a1+(x0+.2)*b1 endloop loop x0:=0,30,11 draw x0,140;x0+8,140 endloop font zsans,20 at 35,130 }}\tim{write }least squares fit\tim{ }\flushleft{\tim{font unit userchoose back bib mode xor text }}\tim{ }Now that you have seen the material once, you may wish to reduce the number of sample points so that you can observe a large number of samples in a shorter time. In addition, you may wish to select a different x value for x at which to test condfidence and prediction limits. To change these parameters, click in the box marked parameters. If you are interested in changing this module's program code, click in the program code box to see information on the formulas used to compute the least squares fit, confidence limits, and prediction limits and information on variables used in the program. If you would like to run the module again, click in the restart box.\tim{ }\flushleft{\tim{\\ * boxes!!! loop i:=0,3 box 3+125*i,375;125*(i+1)-6,396;3 endloop at 8,372 }}\tim{write }Parameters\tim{ }\flushleft{\tim{at 147,372 }}\tim{write }Program\tim{ }\flushleft{\tim{at 275,372 }}\tim{write }Restart\tim{ }\flushleft{\tim{at 415,372 }}\tim{write }Quit loop pause keys=touch if (ztouchy>374)&(ztouchy<397)\tim{ }\flushleft{\tim{ if (ztouchx>2)&(ztouchx<120) box 3,375;119,396;-16 pause 1 jump parameters elseif (ztouchx>127)&(ztouchx<245) box 128,375;244,396;-16 pause 1 jump programcode elseif (ztouchx>252)&(ztouchx<370) box 253,375;369,396;-16 pause 1 jump cover elseif (ztouchx>377)&(ztouchx<495) box 378,375;494,396;-16 pause 1 jump quit endif endif endloop unit quit unit parameters mode xor write To change a parameter, click in the appropriate box. loop i:=1,4 loop j:=0,1 box i*60+160,j*30+75;i*60+210,j*30+96;3 endloop loop j:=5,6 box i*60+160,j*30+75;i*60+210,j*30+96;3 endloop endloop at 0,85 write Number of sample points at 0,235 write Test point for limits at 240,73 write 3 at 300,73 write 5 at 360,73 write 7 at 415,73 write 10 at 235,103 write 15 at 295,103 write 25 at 355,103 write 50 at 410,103 write 100 loop i:=1,4 at i*60+163,223 write x = <|s,i|> at i*60+163,253 write x = <|s,i+4|> endloop }}\tim{box 25,375;475,396;3 text 1,372 }\center{\tim{Click to return to supplemental information page }}\tim{\\ }\flushleft{\tim{loop erase 220,135;500,160 erase 220,285;500,310 at 220,135 write Current setting: n = <|s,n|> at 220,285 write Current setting: x = <|s,xcalc|> calc flag:=0 loop pause keys=touch if (ztouchy>74)&(ztouchy<97) if (ztouchx>219)&(ztouchx<271) calc n:=3 outloop elseif (ztouchx>279)&(ztouchx<331) calc n:=5 outloop elseif (ztouchx>339)&(ztouchx<391) calc n:=7 outloop elseif (ztouchx>399)&(ztouchx<451) calc n:=10 outloop endif elseif (ztouchy>104)&(ztouchy<127) if (ztouchx>219)&(ztouchx<271) calc n:=15 outloop elseif (ztouchx>279)&(ztouchx<331) calc n:=25 outloop elseif (ztouchx>339)&(ztouchx<391) calc n:=50 outloop elseif (ztouchx>399)&(ztouchx<451) calc n:=100 outloop endif elseif (ztouchy>224)&(ztouchy<247) if (ztouchx>219)&(ztouchx<271) calc xcalc:=1 outloop elseif (ztouchx>279)&(ztouchx<331) calc xcalc:=2 outloop elseif (ztouchx>339)&(ztouchx<391) calc xcalc:=3 outloop elseif (ztouchx>399)&(ztouchx<451) calc xcalc:=4 outloop endif elseif (ztouchy>254)&(ztouchy<277) if (ztouchx>219)&(ztouchx<271) calc xcalc:=5 outloop elseif (ztouchx>279)&(ztouchx<331) calc xcalc:=6 outloop elseif (ztouchx>339)&(ztouchx<391) calc xcalc:=7 outloop elseif (ztouchx>399)&(ztouchx<451) calc xcalc:=8 outloop endif elseif (ztouchx>24)&(ztouchx<476)&(ztouchy>374)&(ztouchy<397) calc flag:=1 outloop endif endloop if flag=1 outloop endif endloop box 25,375;475,396;-12 pause 1 jump userchoose unit sample3 $$NEW example of confidence limits for the slope of the population next pg4half back pg4 mode xor do graph text 0,300 }}In the following few screens, you will see an example of confidence limits for the slope of the regression line of the population. Click to see a sample of points. \\\tim{ }\flushleft{\tim{pause keys=touch do berase do points do exprelim do berase at 0,300 }}\tim{write }Now click to see if the <|s,cc|>% confidence limits for the slope include the slope of the true mean.\tim{ }\flushleft{\tim{pause keys=touch loop x0:=0,30,3 dot x0,180 endloop font zsans,20 at 35,170 }}\tim{write }confidence limits\tim{ }\flushleft{\tim{font calc flag:=0 calc clmp:=b1+studT*sdyx/sdx*(n-1)^-.5 clmp1:=b1-studT*sdyx/sdx*(n-1)^-.5 loop x0:=0,8,.1 gdot x0,clmp*x0+a gdot x0,clmp1*x0+a endloop if (clmpb) calc flag:=1 endif do berase at 0,300 if flag=1 }}\tim{ }write The confidence limits do not include the true slope! else write The confidence limits do include the true slope! endif\tim{ }\flushleft{\tim{pause 2 at 0,350 }}\tim{write }Please click to continue.\tim{ }\flushleft{\tim{pause keys=all,touch erase 0,0;150,400 do berase text 0,300 }}Now we will test the confidence limits for slopes of other samples. Click to indicate how many samples you wish to see. \\\tim{ }\flushleft{\tim{do boxes loop l:=1,j do sample3more if flag=1 calc sf(1):=sf(1)+1 else calc sf(2):=sf(2)+1 endif erase 110,0;150,70 at 110,0 show sf(2) at 110,40 show sf(1) endloop calc sf(3):=100*sf(2)/(sf(1)+sf(2)) do berase at 0,300 }}\tim{write }For these samplings, there were <|s,sf(2)|> successes and <|s,sf(1)|> failures for a success rate of <|s,sf(3)|>%.\tim{ }\flushleft{\tim{pause 1 do nextscr jump pg4half }}\enddata{text,204354}